Question

stop & jot
write all the expressions equivalent to $5^8$.
state the property that justifies why the
expressions are equivalent.
a $5^4 \cdot 5^2$
b $(5^4)^2$
c $\frac{5^9}{5}$
d $5^0 \cdot (5^2)^6$
e $\frac{5^{-3}}{5^{-13}}$
f $5^7 \cdot 5$
your work here

Explanation:

Step1: Evaluate Option A (Product Rule)

Product of powers: $a^m \cdot a^n = a^{m+n}$
$5^4 \cdot 5^2 = 5^{4+2} = 5^6$

Step2: Evaluate Option B (Power of Power)

Power of a power: $(a^m)^n = a^{m \cdot n}$
$(5^4)^2 = 5^{4 \cdot 2} = 5^8$

Step3: Evaluate Option C (Quotient Rule)

Quotient of powers: $\frac{a^m}{a^n} = a^{m-n}$
$\frac{5^9}{5} = 5^{9-1} = 5^8$

Step4: Evaluate Option D (Combine Rules)

First power of a power, then product:
$5^0 \cdot (5^2)^6 = 5^0 \cdot 5^{12} = 5^{0+12} = 5^{12}$

Step5: Evaluate Option E (Negative Exponent Rule)

Negative exponents: $\frac{a^m}{a^n} = a^{m-n}$
$\frac{5^{-3}}{5^{-13}} = 5^{-3 - (-13)} = 5^{10}$

Step6: Evaluate Option F (Product Rule)

Product of powers: $a^m \cdot a^n = a^{m+n}$
$5^7 \cdot 5 = 5^{7+1} = 5^8$

Answer:

• B: $(5^4)^2$, justified by the Power of a Power Property: $(a^m)^n = a^{m \cdot n}$
• C: $\frac{5^9}{5}$, justified by the Quotient of Powers Property: $\frac{a^m}{a^n} = a^{m-n}$
• F: $5^7 \cdot 5$, justified by the Product of Powers Property: $a^m \cdot a^n = a^{m+n}$